Parallel Parking: Optimal Entry and Minimum Slot Dimensions

Jiri Vlasak

1,2 a

, Michal Sojka

2 b

and Zden

ek Hanz

alek

2 c

Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic

Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, Czech Republic

Keywords:

Automated Parking, Parallel Parking.

Abstract:

The problem of path planning for automated parking is usually presented as ﬁnding a collision-free path from

initial to goal positions, where three out of four parking slot edges represent obstacles. We rethink the path

planning problem for parallel parking by decomposing it into two independent parts. The topic of this paper is

ﬁnding optimal parking slot entry positions. Path planning from initial to entry position is out of scope here.

We show the relation between entry positions, parking slot dimensions, and the number of backward-forward

direction changes. This information can be used as an input to optimize other parts of the automated parking

process.

1 INTRODUCTION

Driver assistance systems experience a great expan-

sion. Proposals for automated driving are topics of

many teams of engineers and academics. One of the

basic building blocks of automated driving is the (au-

tomated) parking assistant.

A common approach to the problem of path plan-

ning for parallel parking is to geometrically com-

pute a path from the current (initial) position to

the goal position inside the parking slot. When an

optimization-based approach is used, the time or path

length between initial and goal positions is usually

minimized. But what if the given goal position is not

chosen optimally?

In this paper, we reformulate the path planning

problem of automated parking for a parallel parking

slot into the problem of ﬁnding optimal entry and goal

positions. We solve this problem for the given park-

ing slot and car dimensions using a simulation ap-

proach. A feasible path is computed by simulating

the car movement between these positions.

We show how we ﬁnd the entry positions, the de-

pendency between entry positions, parking slot di-

mensions, and the number of backward-forward di-

rection changes, and how to ﬁnd the minimum dimen-

sions of a parking slot for the given car. The results of

our simulations can be used as parameters in parking

https://orcid.org/0000-0002-6618-8152

https://orcid.org/0000-0002-8738-075X

https://orcid.org/0000-0002-8135-1296

assistant algorithms to quickly decide about the pos-

sibility of parking and simplify the navigation toward

the parking slot.

The rest of the paper is structured as follows.

We deﬁne the parallel parking problem in Section 2.

In Section 3, we describe how we ﬁnd the entry

positions, how entry positions depend on direction

changes, and how we compute the minimum dimen-

sions of a parallel parking slot. In Section 4, we per-

form computational experiments and compare our re-

sults to related works. We conclude the paper in Sec-

tion 5. The source code of our algorithm is publicly

available as free and open source software

1.1 Related Works

(Blackburn, 2009) explains how to compute the mini-

mum length of a parallel parking slot, to which the car

can park in only one maneuver. In this paper, we con-

sider parallel parking slots that could be shorter and

therefore we allow multiple backward-forward direc-

tion changes.

(Zips et al., 2013) introduce two-phased constraint

optimization algorithm to plan the parking maneu-

ver for parallel, perpendicular, and angle parking slot.

During the ﬁrst phase, the algorithm computes a path

from the goal position to a phase switching point,

from which the car can leave the slot. In the second

phase, the algorithm ﬁnds a path from the switching

point to the initial position. In their follow-up work

https://rtime.ciirc.cvut.cz/gitweb/hubacji1/bcar.git

300

Vlasak, J., Sojka, M. and Hanzálek, Z.

Parallel Parking: Optimal Entry and Minimum Slot Dimensions.

DOI: 10.5220/0011045600003191

In Proceedings of the 8th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2022), pages 300-307

ISBN: 978-989-758-573-9; ISSN: 2184-495X

(Zips et al., 2016), they present improvements to the

algorithm for a path planning outside the parking slot.

Our paper differs in the problem deﬁnition. We con-

sider the parallel parking slot – not the goal position –

as the input. Our approach guarantees the optimality

of entry positions as opposed to the phase switching

point.

(Vorobieva et al., 2015) present three algorithms

for geometric parking: (1) The algorithm for parking

in one maneuver computes path from the goal posi-

tion by simulating forward movement with the max-

imum steering. This algorithm works only for long

enough parking slots. (2) Parking in several paral-

lel trials algorithm starts by computing the path from

initial position to some position partially inside the

parking slot with the constraint of the car heading be-

ing parallel to the parking slot heading. Then, the

algorithm continues by simulating forward-backward

moves toward the parking slot with the constraint of

the car heading being parallel to the parking slot head-

ing when the car stops for the direction change. (3)

Parking algorithm called several reversed trials gen-

eralize the algorithm for parking in one maneuver. It

computes the path in the reverse order to the park-

ing process: the algorithm starts from the goal po-

sition and computes the path by simulating forward-

backward moves with the maximum steering until the

car leaves the parking slot.

(Li and Tseng, 2016) propose complete system for

automated parking. Their planning algorithm com-

putes a path for a parallel parking slot from three cir-

cle segments. The initial position is considered a part

of the input to the algorithm and only one (backward)

move is allowed.

The works above describe a geometric planner to

ﬁnd a path between initial and goal position. In this

paper, we also use the geometric approach. How-

ever, we consider entry and goal positions as the re-

sult, which allows us to guarantee that the car can

park from entry positions into the parking slot with

the minimum number of backward-forward direction

changes. Path planning from initial to entry position

is out of scope this paper.

(Li et al., 2016) present dynamic optimization

framework for computing a time-optimal maneuvers

for parallel parking. They compute trajectory from

initial position to any parked position, i.e., a position

of a car when the car is completely inside the parking

slot. Compared to our approach, we do not consider

the time. However, we compute a set of entry posi-

tions and return it as the result.

To solve the path planning problem of parallel

parking for automated vehicle, (Jing et al., 2018)

introduce nonlinear programming optimization that

minimizes multiple objectives such as path length,

distance from the car front to the parking slot front,

and the distance from the car center to the parking slot

center. The initial position of the car is ﬁxed and only

one (backward) move is allowed. In this paper, we

allow multiple backward-forward direction changes,

but we minimize their count.

2 PARALLEL PARKING

PROBLEM

We deﬁne the used terminology and the parallel park-

ing problem in this section.

2.1 Deﬁnitions

Car position is a tuple C = (x, y,θ,s,φ), where x and

y are cartesian coordinates of the rear axle center, θ is

car heading, s ∈

{

−1,+1

}

is direction of the move-

ment (backward and forward respectively), and φ ∈

[−φ

max

,+φ

max

] is car steering angle. Car positions

are subject to a discrete kinematic model (Kuwata

et al., 2009) C

k+1

= f (C

), k ∈ N where function f

is given by Eq. (1):

k+1

= x

+ s

· ∆ ·cos(θ

)

k+1

= y

+ s

· ∆ ·sin(θ

)

k+1

= θ

· ∆

·tan(φ

(1)

where ∆ ∈ R

is a positive constant we call the

step distance.

The direction change is a change of the car move-

ment direction s. We denote Γ the number of direction

changes.

Car dimensions is a tuple D = (w,d

,b,φ

max

where w is the width of the car, d

and d

is the dis-

tance from the rear axle center to the front, respective

back of the car, b is the wheelbase (distance between

the front and rear axle), and φ

max

is the maximum

steering angle.

Car frame F (C) is a rectangle given by car posi-

tion C and car dimensions D.

Minimum turning radius is the radius r of the cir-

cle traced by the rear axle center when the car moves

with the maximum steering. It holds that r =

tanφ

max

Curb-to-curb distance is the diameter d of the

circle traced by the outer front wheel when the car

moves with the maximum steering. It holds that

d = 2 ·



r +



+ b

Parallel Parking: Optimal Entry and Minimum Slot Dimensions

301

Figure 1: Orange rectangles represent an example subset

of computed entry positions, green rectangles are goal po-

sitions, and blue frame is parallel parking slot. The cross

represents the middle of the rear axle.

Figure 2: Orange rectangles represent an example subset of

possible entry positions, blue frame is parallel parking slot.

The cross represents the middle of the rear axle.

Parallel parking slot is a rectangle whose one side

(that we call entry side) is adjacent to the road. Paral-

lel parking slot is deﬁned as a tuple P = (p,δ,W,L),

where point p ∈ R

is one corner of the rectangle on

the entry side, δ is the direction vector of the en-

try side relative to p, W > w is parking slot width

greater than the width of the car, and the length of

the parking slot, equal to the length of the entry side,

is L ≥ d

+ d

Goal position is a car position C

for which

F (C

) is completely inside the parallel parking slot

P. We can see an example of different green goal po-

sitions C

in Fig. 1.

Possible entry position for parallel parking slot

positioned on the right side of a road (for left-sided

parking slot the deﬁnition would be symmetric) is a

car position C

= (x,y, θ,−1,+φ

max

) with the follow-

ing properties: (i) right front corner of the car frame

F (C

) corresponds to the parking slot corner p, (ii)

the car heading is in between the angle parallel to the

entry side up to the angle perpendicular to the entry

side, i.e., δ + π ≤ θ ≤ δ + 3/2π, and (iii) car frame

F (C

) does not intersect with non-entry sides of the

parking slot. We denote C

a set of possible entry

positions and we can see an example subset of C

Fig. 2.

2.2 Problem Statement

The main problem we solve is:

1. Given a parallel parking slot and car dimensions,

ﬁnd a set of entry positions C

⊂ C

from which

the car can park into the slot with the minimum

number of direction changes.

There are two additional related problems we solve

along with the main one:

2. Given a parallel parking slot, car dimensions, and

the maximum number of direction changes, what

is a set of all entry positions C

⊂ C

from which

the car can park into the slot?

3. Given the car dimensions and the maximum num-

ber of direction changes, what are the minimum

dimensions (i.e., width and length) of a parallel

parking slot the car can park into?

3 PARKING SIMULATION

ALGORITHMS

In this section, we describe our parallel parking sim-

ulation algorithm, and we show how we use it to

solve the problems from Section 2.2. In Section 3.1,

we describe our algorithm for ﬁnding entry positions

from which it is possible to park into the slot with the

minimum number of direction changes, i.e., the main

problem. In Section 3.2, we relax the condition on

the minimum number of direction changes (problem

2), and in Section 3.3 we describe how we use the al-

gorithm to ﬁnd the minimum dimensions of a parallel

parking slot (problem 3).

3.1 Finding Entry Positions

Our simulation algorithm ﬁrst computes the set of

possible entry positions C

. Then, the algorithm sim-

ulates backward-forward moves from each possible

entry position C

∈ C

until the car frame is com-

pletely inside the parking slot which mean the goal

position is found. We can see the pseudocode in Al-

gorithm 1. ∆

is the increment to the car position

heading for the consequent possible entry positions.

VEHITS 2022 - 8th International Conference on Vehicle Technology and Intelligent Transport Systems

302

Algorithm 1: Finding entry positions.

• Input: car dimensions D, parking slot P

• Output: entry positions C

1: C

← {C

(D,P,θ)|θ = δ + π + m · ∆

,m ∈ N}

2: H ← {(C

,s,φ, Γ)|C

∈ C

, s = −1, φ =

max

,Γ = 0}

3: Γ

max

← ∞

4: I ←

0  Set of goal and entry positions

5: while I =

0 do

6: for (C,C

,s,φ, Γ) ∈ H do

7: if Γ > Γ

max

then

8: continue

9: end if

10: while F(C) not intersects with P do

11: C ← f (C,s, φ)  Simulate move

12: end while

13: if F(C) inside P then

14: I ← I ∪ (C,C

)

15: Γ

max

← Γ

16: else

17: H ← H ∪ (C,C

,−s,−φ, Γ+ 1)

18: end if

19: end for

20: end while

21: return C

= {C

|(C,C

) ∈ I}

3.2 Limit on the Direction Changes

Algorithm 1 ﬁnds the set of entry positions from

which it is possible to park into the parking slot with

the minimum number of direction changes. To ﬁnd

all entry positions C

⊂ C

for the given maximum

number of direction changes, we provide Γ

max

as the

input to the Algorithm 1 and remove Γ

max

update in

Line 15.

By removing Γ

max

update, the computed set of en-

try positions does not need to be continuous, i.e., the

difference in headings of neighboring entry positions

could be greater than ∆

. We can see such a situation

in Fig. 1, where orange rectangles represent an ex-

ample subset of computed entry positions where the

continuity is broken approximately in the middle.

3.3 Minimum Slot Dimensions

To ﬁnd the minimum dimensions (i.e., width and

length) of a parallel parking slot for the given car di-

mensions and maximum number of direction changes

max

, we run Algorithm 1 for ﬁnding entry positions

repeatedly, looping over different parking slot widths

and lengths. First, we initialize the parking slot width

W = w and length L = d

to match the car dimen-

sions. Then, we gradually increase the parking slot

-1

-6 -4 -2 0 2 4

Car width w [m]

Car length d

+ d

[m]

"mid-sized vehicle"

Renault ZOE

Opel Corsa

Volkswagen transporter

Mercede-Benz AMG GT

Figure 3: Car frame and the simulated path given by the

forward movement with the maximum steering angle for

“mid-sized vehicle”, Renault ZOE, Opel Corsa, Volkswa-

gen transporter, and Mercedes-Benz AMG GT.

width and length by the width step ∆

and the length

step ∆

respectively.

4 RESULTS

In this section, we present the results of the com-

putational experiments. In Section 4.1, we compare

our approach to the related works. In Section 4.2,

we show the range of all entry positions heading for

the given maximum number of direction changes. Fi-

nally, we show the minimum dimensions of a parallel

parking slot in Section 4.3.

We provide the results for ﬁve different cars. Their

dimensions are shown in Table 1. In Fig. 3, we can see

the car frames and the simulated paths given by the

forward movement with the maximum steering angle.

The width of the car is always without left and right

rear view mirrors.

4.1 Comparison to Related Works

(Zips et al., 2013) use the dimensions of the “mid-

sized vehicle” and the parallel parking slot with the

width W = 2.2 m and the length L = 5.1 m for their

simulations. There is no change to the ﬁrst phase of

their algorithm in their follow-up work.

Along with the parallel parking slot and car di-

mensions, (Zips et al., 2013) are given the initial and

goal positions. There is no further information or ex-

act values provided in their simulation studies. They

need at least 12 direction changes to park in the slot.

We compute that it’s possible to park into the slot

they use with 10 direction changes for the given car

dimensions, so we conclude that the used goal posi-

tion is not optimal.

(Vorobieva et al., 2015) use the dimensions of the

Renault ZOE for their experiments. They use several

Parallel Parking: Optimal Entry and Minimum Slot Dimensions

303

Table 1: Car dimensions used in the computations.

Car w [m] d

[m] d

[m] b [m] φ

max

[

◦

]

“mid-sized vehicle” 1.8 3.7 1.0 2.7 45

Renault ZOE 1.771 3.427 0.657 2.588 33

Opel Corsa 1.532 3.212 0.410 2.343 32

Volkswagen transporter 1.994 4.308 1.192 3.400 36

Mercedes-Benz AMG GT 1.939 3.528 1.016 2.630 32

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8

Number of direction changes [-]

Parallel parking slot length [m]

Vorobieva et al., 2015

this paper

Figure 4: Comparison of direction changes for Renault

ZOE when several reversed trials method is used with goal

positions from (Vorobieva et al., 2015) or with goal posi-

tions from this paper. Goal positions from (Vorobieva et al.,

2015) has the car frame aligned with the entry and rear sides

of the parallel parking slot. The parallel parking slot has the

width W = 2.0 m and variable length of up to L = 5.8 m with

the length step of ∆

= 0.01m.

reversed trials method to plan a path into the parallel

parking slot.

We compare our implementation of several re-

versed trials for goal positions used in (Vorobieva

et al., 2015) with goal positions computed by our ap-

proach.

The goal position used in (Vorobieva et al., 2015)

is not explicitly stated. They assume the goal position

is known. For our computational experiments, we de-

duce the goal position from the context of the paper,

i.e. the goal position for parallel parking slot posi-

tioned on the right side of a road has the following

properties: (i) left side of the car frame given by the

goal position corresponds to parking slot entry side,

and (ii) rear side of the car frame given by the goal

position corresponds to parking slot rear side.

The parallel parking slot has the width W = 2.0m,

and we test the length of up to L = 5.8 m with the

length step of ∆

= 0.01 m.

In Fig. 4, we can see that when goal position

is given by our approach, it leads to less direction

changes.

(Li et al., 2016) use the dimensions of the Renault

ZOE for their simulations. The parallel parking slot

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Heading of entry positions [°]

Extra slot width W - w [m]

max

=10

max

Figure 5: Entry positions heading for Renault ZOE, the par-

allel parking slot with different extra slot width W − w with

the width step of ∆

= 0.01m, constant extra slot length

L − d

− d

= 1.1 m, and different maximum number of di-

rection changes Γ

max

has the width W = 2.0m and the length L = 4.5 m.

(Li et al., 2016) need 19 direction changes for the

given parallel parking slot and car dimensions. Our

approach results in 19 direction changes for the given

values, too.

4.2 Limit on the Direction Changes

To compute the set of all entry positions for the given

maximum number of direction changes, we use the

car dimensions of the Renault ZOE.

In Fig. 5, we can see the entry positions heading

for the parallel parking slot with constant extra slot

length L − d

− d

= 1.1 m and different maximum

number of direction changes Γ

max

= 1, 2, 5, and 10

respectively.

We can see that the range of entry positions head-

ing is wider for larger extra slot width W − w and for

increasing Γ

max

. Also, we can see that the entry po-

sitions are found for narrower slots for higher Γ

max

Finally, the set of entry positions does not need to

be continuous, i.e., the difference between two neigh-

boring entry positions could be greater than ∆

. The

range of entry positions heading then consists of the

subranges.

In Fig. 6, we can see the number of entry positions

VEHITS 2022 - 8th International Conference on Vehicle Technology and Intelligent Transport Systems

304

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of subranges [-]

Extra slot width W - w [m]

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Figure 6: The number of entry positions heading subranges

for Renault ZOE, the parallel parking slot with different ex-

tra slot width W − w with the width step of ∆

= 0.01 m,

extra slot length L − d

− d

from 0.5m up to 1.9m with the

length step of ∆

= 0.1m, and constant Γ

max

= 10.

heading subranges for the parallel parking slot with

different extra slot lengths L − d

− d

from 0.5m up

to 1.9m and constant Γ

max

= 10.

We can see that there is only one range of entry po-

sitions heading for wide enough slots. The number of

subranges increases for narrower slots but decreases

again for the slots that are too narrow.

4.3 Minimum Slot Dimensions

To compute the minimum dimensions of a parallel

parking slot, we use the possible entry position head-

ing step ∆

= 10

−4

rad, parking slot width step ∆

0.01m, and the length step ∆

= 0.01 m.

In Fig. 7, we can see the computed minimum di-

mensions of a parallel parking slot for Renault ZOE

and different maximum number of direction changes

max

Renault ZOE needs to perform at least Γ = 30 di-

rection changes when parking into the parallel park-

ing slot with extra slot length L − d

− d

= 0.4 m.

Also, there is a negligible improvement of 0.02m

when increasing the number of direction changes to

Γ = 100.

In Fig. 8, we can see the minimum dimensions of

a parallel parking slot for Opel Corsa, Volkswagen

transporter, and Mercedes-Benz AMG GT.

Opel Corsa can park into the parallel parking slot

with extra slot length of 0.4 m with Γ = 10 direc-

tion changes. Volkswagen transporter needs the same

number of direction changes Γ = 30 as the Renault

ZOE, although it requires more extra slot width. It

is not possible to park Mercedes-Benz AMG GT into

the parallel parking slot with extra slot length of 0.4m

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5

Extra slot length L - d

- d

[m]

Extra slot width W - w [m]

Renault ZOE

max

=10

max

=20

max

=30

max

=40

max

=50

max

=100

Figure 7: Computed minimum dimensions of a parallel

parking slot for Renault ZOE and different maximum num-

ber of direction changes. The ﬁrst line at the top of the graph

shows the minimum dimensions when no direction change

is allowed (Γ

max

= 0), followed by the maximum number

(Γ

max

) of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, and

100 direction changes. The width step ∆

= 0.01 m and the

length step ∆

= 0.01m.

within the reasonable number of backward-forward

direction changes.

The improvement to extra slot length for Γ > 20

is negligible, i.e., in the order of 10

−2

m. The shape

of the slot’s minimum dimensions curve depends on

whether the number of backward-forward direction

changes is even or odd.

Fig. 9 shows the minimum dimensions of a paral-

lel parking slot for all the cars. The maximum number

of direction changes shown in the ﬁgure is Γ

max

= 1,

2, 5, and 10 respectively.

5 CONCLUSION

In this paper, we re-formulate the path planning prob-

lem for parallel parking: Given a parallel parking slot

and car dimensions, we compute a set of entry po-

sitions to the parking slot from which it is possible

to park into the slot with the minimum number of

backward-forward direction changes. We also use our

approach to compute the minimum dimensions (i.e.,

width and length) of a parallel parking slot for the

given car dimensions.

Our algorithm is designed for ofﬂine use. By pre-

computing entry positions for different parallel park-

ing slots and car dimensions and using them in online

planning, we can speed up the path planning as it does

not need to plan the path inside the parking slot. Park-

ing assistants using our results can ensure that when

the automated vehicle reaches one of the entry posi-

Parallel Parking: Optimal Entry and Minimum Slot Dimensions

305

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5

Extra slot length L - d

- d

[m]

Extra slot width W - w [m]

Opel Corsa

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5

Extra slot length L - d

- d

[m]

Extra slot width W - w [m]

Volkswagen transporter

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5

Extra slot length L - d

- d

[m]

Extra slot width W - w [m]

Mercedes-Benz AMG GT

Figure 8: Computed minimum dimensions of a paral-

lel parking slot for Opel Corsa, Volkswagen transporter,

Mercedes-Benz AMG GT, and different maximum number

of direction changes.

tions, it can optimally park into the parallel parking

slot.

The range of entry positions from which the given

car can park optimally depends on the slot dimen-

sions. The range is wider for larger slots and a higher

number of backward-forward direction changes as

shown in Section 4.

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5

Extra slot length L - d

- d

[m]

Extra slot width W - w [m]

Renault ZOE

Opel Corsa

Volkswagen transporter

Mercede-Benz AMG GT

Figure 9: Computed minimum dimensions of a parallel

parking slot for Renault ZOE, Opel Corsa, Volkswagen

transporter, and Mercedes-Benz AMG GT. The width step

∆

= 0.01 m and the length step ∆

= 0.01 m. The maxi-

mum number of direction changes is Γ

max

= 1, 2, 5, and 10

respectively.

By precomputing minimum dimensions of differ-

ent parallel parking slots, we can simplify the decision

of whether it is possible to park into the slot.

Using more than 20 direction changes does not

lead to signiﬁcant reduction of the parking slot size.

The shape of the slot’s minimum dimensions curve

depends on whether the number of backward-forward

direction changes is even or odd.

ACKNOWLEDGEMENTS

Research leading to these results has received fund-

ing from the EU ECSEL Joint Undertaking and the

Ministry of Education of the Czech Republic un-

der grant agreement 826452, 8A19011, and MSMT-

24623/2019-4/11 (project Arrowhead Tools).

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